Black Scholes Calculator
Price a European call or put with the Black-Scholes-Merton model. Enter spot and strike in dollars, time to expiration in years, volatility and interest rate in percent, and an optional continuous dividend yield — you get the option price, delta, and d1/d2.
Example: with Option type Call · Spot price S ($) 100 · Strike price K ($) 100 · Time to expiration (years) 1 · Volatility σ (%/yr) 20 → Option price: $10.45 per share ($1,045 per 100-share contract).
- Delta0.637 (call gains ~$0.64 per $1 move up)
- d1 and d2d1 = 0.3500, d2 = 0.1500
Computed by the calculator below using its default values. Change any input to see your own numbers.
Call = S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2). N(x) is computed via the Abramowitz-Stegun 7.1.26 erf approximation (max error 1.5×10⁻⁷).
What the Black-Scholes model does
Black-Scholes prices a European option — exercisable only at expiration — from five inputs: spot price, strike, time, volatility, and the risk-free rate (plus a dividend yield in the Merton extension). The core idea is no-arbitrage: a continuously rebalanced mix of stock and cash can replicate the option's payoff, so the option must cost what that hedge costs. The famous d1 and d2 terms measure, roughly, how many standard deviations the stock sits from the strike; N(d2) is the risk-neutral probability the option finishes in the money.
Volatility is the only input you cannot look up — and the price is extremely sensitive to it. Trading desks run the model backward, solving for the implied volatility that matches a market price. This calculator evaluates the normal CDF with the Abramowitz-Stegun 7.1.26 approximation, accurate to about 1.5×10⁻⁷, so results match textbook tables to the cent: Hull's standard example (S=$42, K=$40, σ=20%, r=10%, T=0.5) prices the call at $4.76 and the put at $0.81.
Where the model bends
The assumptions are strong: constant volatility, lognormal prices, no early exercise, continuous frictionless hedging. Real markets show volatility smiles, jumps, and American-style exercise on most single-stock options. Treat the output as a disciplined baseline — for American puts or dividend-heavy calls near ex-dates, binomial models fit better.
How it’s calculated
d1 = [ln(S/K) + (r − q + σ²/2)T] ÷ (σ√T); d2 = d1 − σ√T. Call = S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2); Put = K·e^(−rT)·N(−d2) − S·e^(−qT)·N(−d1). Delta = e^(−qT)·N(d1) for calls, e^(−qT)·(N(d1)−1) for puts. N(·) uses erf via Abramowitz & Stegun eq. 7.1.26 (|error| ≤ 1.5×10⁻⁷). Volatility, r, and q are annual; T in years.
European exercise, constant σ and r, lognormal prices, no transaction costs — American-style options and volatility skew push real prices away from the model.
Call value by strike (S = $100, σ = 20%, r = 5%, T = 1 yr, q = 0)
| Strike | Moneyness | Black-Scholes call |
|---|---|---|
| $90 | In the money | $16.70 |
| $100 | At the money | $10.45 |
| $110 | Out of the money | $6.04 |
| $120 | Far out of the money | $3.25 |
Computed with the Black-Scholes formula above; rounded to cents.
Common mistakes
- Entering time in days or months — T is in years, so 6 months is 0.5 and 30 days is about 0.082.
- Typing volatility as a decimal (0.20) when the field expects percent (20), which collapses the price toward intrinsic value.
- Pricing American options with it: early exercise (especially deep puts and calls before ex-dividend) makes them worth more than Black-Scholes says.
- Forgetting dividends — for a dividend payer, q = 0 overprices calls and underprices puts.
Frequently asked questions
What is the Black-Scholes formula?
Call = S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2), with d1 = [ln(S/K) + (r − q + σ²/2)T]/(σ√T) and d2 = d1 − σ√T. N is the standard normal CDF; the put follows by the same terms with signs flipped.
How is the normal CDF computed here?
Via the error function using the Abramowitz-Stegun 7.1.26 polynomial approximation, N(x) = ½[1 + erf(x/√2)], accurate to about 1.5×10⁻⁷ — well past penny precision for option prices.
Why does my broker show a different price?
Market prices embed implied volatility, bid-ask spreads, and American-style early exercise. If you input historical volatility, expect gaps; feed in the option's implied volatility and prices converge.
What do d1, d2, and delta mean?
N(d2) is the risk-neutral probability the option expires in the money; N(d1) additionally weights how deep. Delta (e^(−qT)N(d1) for calls) estimates the price change per $1 move in the stock and doubles as a hedge ratio.
Does this work for puts and dividend stocks?
Yes — choose Put, and enter a continuous dividend yield q. The Merton adjustment discounts the stock leg by e^(−qT), lowering call values and raising put values versus the no-dividend case.